Question

In a quadrilateral, the angles
6.
The angles of a quadrilateral cannot be in the ratio 1:2:3:6. Why? Give reason
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Answer 1

Answer :

The angles of a quadrilateral cannot be in the ratio 1:2:3:6

Solution :

First, let's assume the angles by multiplying the ratios with a constant term i.e., x

Let the angles of the quadrilateral be

• x
• 2x
• 3x and
• 6x

Here, we'll use the concept :

Sum of all the interior angles of the quadrilateral = 360°

 x + 2x + 3x + 6x = 360°

  12x = 360°

     x = 360°/12

     x = 30°

The value of x is 30°

Finding the angles :-

x = 30°

2x = 2(30°) = 60°

3x = 3(30°) = 90°

6x = 6(30°) = 180°

Therefore, the angles of the quadrilateral are 30° , 60°, 90° and 180°

But, we can not draw a quadrilateral with one of it's angles = 180°

A quadrilateral has four sides, four vertices and four angles.

Any angle of a quadrilateral must be less than 180°

If one angle measures 180°, 3 of the 4 vertices will be collinear which results in a polygon having 3 sides. (Triangle)

 $\setlength{\unitlength}{1 cm}\begin{picture}(0,0)\thicklines\qbezier(1, 0)(1,0)(3,3)\qbezier(5,0)(5,0)(3,3)\qbezier(5,0)(1,0)(1,0)\put(3,0){\circle*{0.1}}\qbezier(2.6,0)(3,0.5)(3.4,0)\put(2.7,0.4){\sf 180^{\circ} }\qbezier(1.3,0.3)(1.5,0.2)(1.3,0)\put(1.5,0.2){\sf 30^{\circ} }\qbezier(4.7,0)(4.3,0.3)(4.8,0.3)\put(4.1,0.2){\sf 60^{\circ} }\qbezier(2.7,2.5)(3,2.1)(3.3,2.5)\put(2.8,2){\sf 90^{\circ} }\end{picture}$

So, The angles of a quadrilateral cannot be in the ratio 1:2:3:6

Answer 2

$\huge\sf\underbrace{Solution}$

Given ratio 1:2:3:6

So,let the common multipluer be x

Angles are 1x, 2x , 3x , 6x

Their sum should be equal to 360°

1x + 2x + 3x + 6x = 180°

12x = 180°

x = 15° So,

x = 15

2x = 30°

3x = 45°

6x = 180°

And the quadrilateral doesnot form Because Angle sum is not equal to 360°

15° + 30° + 45° + 180° = 270°

So,its sum is not equal to 360°

and another reason is

angles of quadrilateral must less than 180°

But The angle formed is equal to 180°

So the quadrilateral doesnot form

So,the angles of quadrilateral doesnot form in ratio of 1:2:3:6